It's fundamentally not Mill's approach. As a utilitarian, Mill got to dodge questions like "what is human flourishing?" Nussbaum, I believe, refers to her interpretation of virtue ethics to answer that question and give CA ultimate grounding.

I'd be wary of quoting that article like it's some authority. At a glance, it reads as skeptical of Mill. The harm principle is the basis of his thinking and he was very concerned about paternalism. I think it's quite likely he would view CA as an overreach.